The electron-doping-driven collapse of the charge gap and staggered magnetization of the spin-orbit-assisted Mott insulator Sr$_{3}$Ir$_{2}$O$_{7}$ is explored via first-principles computational methods. In the antiferromagnetic phase, the gap and magnetization are observed to decrease slowly with increasing doping, with an abrupt collapse of both the gap and the magnetization at an electron concentration corresponding to 4.8% substitution of Sr with La, in excellent agreement with experiment. Additionally, we describe the structural effects of electron doping in Sr$_{3}$Ir$_{2}$O$_{7}$ via a competition between the steric effect from smaller La atoms substituted within the lattice and the dominant doping-driven deformation-potential effect. Curiously, our first-principles calculations fail to capture the low-temperature structural distortion reported in the low-gap phase of Sr$_{3}$Ir$_{2}$O$_{7}$, supporting the notion that this distortion arises as a secondary manifestation of an unconventional electronic order parameter in this material.
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